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@@ -108,7 +108,7 @@ As with all approximate inference methods, the performance of the LA depends on
 
 # $\textcircled{4}$ Approximate Predictive Distribution
 
-To predict using a posterior (approximation) $p ( \theta \mid \mathcal { D } )$ , we need to compute $p ( y \mid f ( x _ { * } ) , \mathcal { D } ) \ =$ $\begin{array} { r } { \int p \dot { ( \boldsymbol { y } \vert } f _ { \boldsymbol { \theta } } ( x _ { * } ) ) \dot { p ( \boldsymbol { \theta } \vert } \dot { \mathcal { D } } ) d \boldsymbol { \theta } } \end{array}$ for any test point $x _ { * } \in \mathbb { R } ^ { n }$ , which is intractable in general. The sim| ⇤ D       (✓s)Ss=1 from p(✓ | D): p(y | f (x⇤), D) ⇡ S1 PSs=1 p(y | f✓s (x⇤)). However, for LAs with GGN [26] attribute this to the inconsistency between Hessian approximation and the predictive and suggest to use a linearized predictive instead, which can also be useful for theoretic analyses [28]. For the last-layer LA, the Hessian coincides with the GGN and the linearized predictive is exact.
+To predict using a posterior (approximation) $p ( \theta \mid \mathcal { D } )$ , we need to compute $p ( y \mid f ( x _ { * } ) , \mathcal { D } ) \ =$ $\begin{array} { r } { \int p \dot { ( \boldsymbol { y } \vert } f _ { \boldsymbol { \theta } } ( x _ { * } ) ) \dot { p ( \boldsymbol { \theta } \vert } \dot { \mathcal { D } } ) d \boldsymbol { \theta } } \end{array}$ for any test point $x _ { * } \in \mathbb { R } ^ { n }$ , which is intractable in general. The sim| ⇤ D       (✓s)Ss=1 from p(✓ | D): p(y | f (x⇤), D) ⇡ S1 PSs=1 p(y | f✓s (x⇤)). However, for LAs with GGN [26] attribute this to the inconsistency between Hessian approximation and the predictive and suggest to use a linearized predictive instead, which can also be useful for theoretic analyses [28]. For the last-layer LA, the Hessian coincides with the GGN and the linearized predictive is exact.
 
 The predictive of a linearized neural network with a LA approximation to the posterior $p ( \boldsymbol { \theta } | \mathcal { D } ) \approx$ $\mathcal { N } ( \theta ; \theta _ { \mathrm { M A P } } , \mathcal { \Sigma } )$ results in a Gaussian distribution on neural network outputs $f _ { * } : = f ( x _ { * } )$ and therefore enables simple approximations or even a closed-form solution. The distribution on the outputs is given by $\bar { p } ( f _ { * } | \bar { x } _ { * } , \mathcal { D } ) \approx \mathcal { N } ( f _ { * } ; f _ { \theta _ { \mathrm { M A P } } } ( x _ { * } ) , J ( x _ { * } ) ^ { \intercal } \varSigma J ( x _ { * } ) )$ and is typically significantly lowerdimensional (number of outputs $C$ instead of parameters $D$ ). It can also be inferred entirely in function space as a Gaussian process [25, 26]. Given the distribution on outputs $f _ { * }$ , the predictive distribution can be obtained by integration against the likelihood: $\begin{array} { r } { p ( y | x _ { * } , \mathcal { D } ) = \int p ( y | \bar { f } _ { * } ) p ( f _ { * } | x _ { * } , \mathcal { D } ) d \theta } \end{array}$ . In the case of regression with a Gaussian likelihood with variance $\sigma ^ { 2 }$ , the solution can even be obtained analytically: $\bar { p } ( y | x _ { * } , \mathcal { D } ) \approx \mathcal { N } ( y ; f _ { \theta _ { \mathrm { M A P } } } ( x _ { * } ) , J ( x _ { * } ) ^ { \intercal } \varSigma J ( x _ { * } ) + \sigma ^ { 2 } I )$ . For non-Gaussian likelihoods, e.g. in classification, a further approximation is needed. Again, the simplest approximation to this is Monte Carlo integration. In the binary case, we can employ the probit approximation [31, 16] which approximates the logistic function with the probit function. In the multi-class case, we can use its generalization, the extended probit approximation [52]. Finally, first proposed for non-BNN applications [53, 54], the Laplace bridge approximates the softmax-Gaussian integral via a Dirichlet distribution [55]. The key advantage is that it yields a distribution of the integral solutions.